Derek Riordan scored twice as Celtic survived an early scare before easing aside Livingston to move into the quarter-finals of the Scottish Cup.
Derek Riordan scored twice for Celtic
Dave Mackay headed the First Division side into an 18th-minute lead.
But Darren O'Dea fired the equaliser and Riordan put Celtic ahead before half-time with a superb drive.
Riordan got in behind the defence to fire in the third and Jan Vennegoor of Hesselink powered home the fourth for the rampant Scottish champions.
Paul Hartley made a quiet debut in the Celtic midfield in place of Thomas Gravesen following his midweek transfer.
But he was not alongside former Hearts team-mate Steven Pressley, who was out with a back injury and replaced with O'Dea.
Livingston boss John Robertson was forced to bring in midfielder Liam Fox for injured on-loan Hearts striker Jamie Mole as he sought the West Lothian side's first win over Celtic.
And the First Division side gave as good as they got before taking the lead after 18 minutes.
Lee Makel floated in a free-kick and Mackay flicked a header into the corner of the net from 16 yards.
Livi fans began to dream that their side could repeat Clyde's feat last season of knocking the Scottish champions out the cup.
But the goal spurred Celtic into action and they equalised after 29 minutes.
Vennegoor of Hesselink rose above the defence to head down a Neil Lennon free-kick and the unmarked O'Dea celebrated his 20th birthday by blasting home from 10 yards.
Makel was lucky to survive a penalty claim when he appeared to trip Shunsuke Nakamura from behind.
But it was Livi who were infuriated by referee Charlie Richmond in the lead-up to Celtic's second goal.
Steve McManus appeared to win the ball fairly from Neil Teggart, but the home side thought that the game should have been stopped as the striker lay injured after the challenge.
Paul Hartley (left) made a quiet debut in Celtic's colours
However, Celtic continued, O'Dea set up Riordan and the former Hibernian forward cleverly stabbed the ball into the far corner from a difficult angle.
Livi started the second half brightly and squandered a great chance when Steven Craig failed to connect with Allan Walker's low cross into the six-yard box.
Celtic did not take long to make them pay for that miss.
Nakamura's neat through ball found Riordan making a clever run behind the defence and the Scotland international clinically drove the ball home.
It was Nakamura who was the provider again less than two minutes later, putting Vennegoor of Hesselink clear and, despite a hint of offside, the striker drove the ball low into the far corner from the edge of the box.
Substitute Gravesen had an effort cleared off the line, but Celtic were already on their way to the next round.
Livingston: Stewart, MacKay, McPake, Tweed, Golabek, Fox (Hamill 57), Mitchell, Walker, Craig (Smylie 82), Makel (Dorrans 63), Teggart.
Subs Not Used: Wight, Miller.
Goals: MacKay 18.
Celtic: Boruc, Telfer, McManus, O'Dea, Naylor, Nakamura, Hartley, Lennon (Sno 76), Jarosik (Gravesen 68), Vennegoor of Hesselink, Riordan.
Subs Not Used: McGovern, Wilson, Beattie.
Goals: O'Dea 30, Riordan 45, 59, Vennegoor of Hesselink 61.
Ref: C Richmond